ZFC-Based Proof of Global Regularity for the 3D Incompressible Navier–Stokes Equations

This document presents a formally structured and mathematically rigorous argument addressing the global existence and smoothness of solutions to the three-dimensional incompressible Navier–Stokes equations. The result is formulated within the theoretical framework of the Millennium Prize Problem posed by the Clay Mathematics Institute (2000), and applies to the standard domains: bounded subsets of Euclidean space (R³), the entire space R³, and the periodic torus T³. The analysis integrates classical methods — including Galerkin approximation, energy estimates, Sobolev interpolation, and nonlinear regularity conditions — with novel mechanisms such as an Extended Self-Similarity (ESS) criterion and logarithmic Sobolev damping. Together, these yield uniform a priori bounds, high-order differential inequalities, and compactness-based convergence sufficient to establish smoothness within the formal framework adopted. Numerical simulations are provided to support key analytic decay estimates and to illustrate suppression of high-frequency modes. The construction is entirely expressible within Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC), and is compatible with mechanized formalization in proof assistants. This version is intended for peer review and scholarly examination. It is released in the spirit of open verification and with full methodological transparency.